3. Notations

We will note by:

\(\mathit{Id}\)	identity matrix
\(\mathit{Tr}A\)	Tensor trace \(A\)
\(\tilde{A}\)	deviatoric part of tensor \(A\) defined by
\(\text{:}\)	doubly contracted product:
	tensor product:
\({A}_{\mathit{eq}}\)	equivalent Von Mises value in the Hill sense defined by
\(M\)	Hill anisotropy matrix
,, E,, K	isotropic elasticity coefficients
\(\alpha\)	coefficient of thermal expansion
\(T\)	temperature
\({T}_{\mathit{ref}}\)	reference temperature

Moreover, in the context of time discretization, all the quantities evaluated at the previous instant are indexed by \(\text{-}\), the quantities evaluated at the instant \(t+\Delta t\) are not indexed and the increments are designated by \(\Delta\). We thus have:

(3.1)\[ \ mathrm {\ Delta} Q=Q- {Q} ^ {\ text {-}}\]